1. Field of the Invention
The present invention relates to an apparatus and method for interpolating sampled signals in which an input signal is sampled and then reproduced to form an output signal using spline interpolation.
2. Background Information
Conventional digital storage oscilloscopes, data recorders, or the like are known to include some form of interpolation apparatus to reproduce an input signal from samples of the input signal.
For example, when a sampled input signal is to be enlarged along the time axis in these apparatus, interpolation data is calculated and inserted between sampled data so that the interpolation data and sampled data can be displayed on a display unit such as a CRT display unit. In such conventional apparatus, linear interpolation, sine interpolation or spline interpolation is used to produce the interpolation data.
Conventional spline interpolation produces a spline curve connecting sampled points. A cubic or third power function curve is commonly used as the spline curve because of processing speed. In this case, a cubic curve that passes two adjacent sampled points is determined, and then a plurality of such cubic curves thus determined are connected to form a spline curve.
Coefficients of the cubic curve that passes two sampled points are determined on the basis of four types of information: sampled data of the two adjacent sampled points and two differential coefficients which satisfy the cubic curve at the two sampled points.
The sampled data of the two sampled points are given already when the sampling of the points is completed, and the differential coefficients at the two sampled points can be obtained as follows:
First, the following five sampled points are selected: a sampled point at which the differential coefficient is to be calculated (central sampled point); two sampled points preceding the central sampled point; and two sampled points following the central sampled point. Second, a biquadratic curve that passes the five sampled points is determined on the basis of the sampled data of these five points. Next, the differential coefficient of the biquadratic curve is calculated at the central sampled point in question, thus determining the differential coefficient at this sampled point. More specifically, a differential coefficient t.sub.0 at the central sampled point is given by the following equation when the y coordinates (i.e., sampled data) of the five sampled points are y.sub.-2, y.sub.-1, y.sub.0, y.sub.1 and y.sub.2 : ##EQU1## Likewise, various differential coefficients at different sampled points can be determined.
By using the differential coefficients thus determined and the sampled data of two adjacent sampled points, a cubic curve that interpolates the two adjacent sampled points can be obtained. By sequentially connecting a plurality of these cubic curves a conventional spline curve is obtained.
When an input signal to be enlarged is inputted, the desired number of interpolation data is first determined according to a magnification factor of the time base enlargement between sampled points. The required interpolation data between each two adjacent sampled points is obtained by using the cubic curves forming the spline curve. Then the interpolation data thus obtained are outputted as display data together with the sampled data of the sampled points.
The conventional apparatus using a spline curve can produce good reproduced signals when the input signal to be sampled changes smoothly.
Electric signals to be displayed on a digital storage oscilloscope, however, include various types of step-like transitions, such as those of pulses. The resultant signals produced from such input signals using conventional apparatus have "undulations" before and after the step-like transitions as illustrated by 1!- 8! in FIG. 1. This presents a problem in that the portions of the reproduced signals corresponding to the steady portions before and after the step-like transition of the input signal are unnaturally variable.